Title Page
Contents
Abstract 11
Ⅰ. Introduction 13
Ⅱ. Analytical thin-walled beam theory based on the Reissner's mixed variational theorem 18
1. Strain-displacement relations 18
2. Laminate constitutive relations 28
2.1. On-axis lamina stress-strain relations 28
2.2. Transformation of stresses and strains 29
2.3. Off-axis lamina stress-strain relations 30
2.4. Laminate stress-strain relations 31
3. Governing equations 36
3.1. Shell equilibrium equations 36
3.2. Displacement univaluedness conditions 43
3.3. Cross-sectional stress resultants 46
3.4. Euler-Bernoulli level stiffness 48
3.5. Timoshenko level stiffness 50
3.6. Timoshenko - Vlasov level stiffness 59
4. Inertial properties 63
5. Section offsets 67
5.1. Shear center 67
5.2. Mass center 67
5.3. Tension center 68
6. Stress recovery 69
Ⅲ. One-dimensional finite element analysis of thin-walled composite beam cross-sections 71
1. Strain-displacement relations 71
2. Finite element approximations 76
3. Governing equations 79
3.1. Decomposition of governing equations 85
3.2. Constraints on warping 87
3.3. Complete set of governing equations 89
3.4. Cross-sectional stiffness matrix 90
Ⅳ. Analysis of rotating thin-walled composite beams based on a geometrically exact beam theory 92
1. Kinematics 93
2. Governing equations 107
2.1. Kinetic energy 107
2.2. Strain energy 109
2.3. Euler-Lagrange equations 110
3. Finite element formulation 113
3.1. Nonlinear static analysis 116
3.2. Normal mode analysis 117
Ⅴ. Numerical examples 120
1. Validation of the geometrically exact beam analysis 120
1.1. Princeton beam experiment 120
2. Validation of cross-section analysis 125
2.1. Isotropic beam 125
2.2. Composite beams 128
Ⅵ. Conclusions and recommendations for future work 158
1. Conclusions 158
2. Recommendations for future work 159
References 160
Abstract (in Korean) 172
Table 5.1. Comparison of section properties for highly heterogeneous section. 127
Table 5.2. Comparison of stiffness constants of single-cell composite box section with CUS layup. 132
Table 5.3. Comparison of rotating natural frequencies (Hz) of single-cell composite box section with CUS layup at Ω=1002 rpm. 136
Table 5.4. Comparison of stiffness constants of single-cell composite box section with CAS layup. 139
Table 5.5. Comparison of rotating natural frequencies (Hz) of single-cell composite box section with CAS layup at Ω=1002 rpm. 142
Table 5.6. Comparison of stiffness constants of the anisotropic I section. 150
Table 5.7. Comparison of stiffness constants of a realistic wind turbine blade section. 156
Figure 1.1. Concept of the cross-section warping. 14
Figure 2.1. Schematic of the reference coordinate frames. 19
Figure 2.2. Shell forces and moments 33
Figure 4.1. Schematic of the beam reference line before and after deformation with reference coordinate frames. 94
Figure 4.2. Schematic of the rotation from xyz frame to 𝜉η𝜁′ frame. 98
Figure 4.3. Schematic of the rotating beam with reference coordinate frames. 103
Figure 5.1. Schematic of the Princeton beam experiment 121
Figure 5.2. Tip chordwise displacement. 122
Figure 5.3. Tip flapwise displacement. 123
Figure 5.4. Tip twist. 124
Figure 5.5. Schematic of highly heterogeneous section. 125
Figure 5.6. Schematic of composite single-cell box section. 129
Figure 5.7. Isometric view of fundamental warping modes of the CUS box section under unit sectional loads. 133
Figure 5.8. Planar view of fundamental warping modes of the CUS box section under unit sectional loads. 134
Figure 5.9. Schematic of shell FE model for single-cell composite box beam. 135
Figure 5.10. Isometric view of fundamental warping modes of the CAS box section under unit sectional loads. 140
Figure 5.11. Isometric view of fundamental warping modes of the CAS box section under unit sectional loads. 141
Figure 5.12. Fan diagram of the CAS box section (ply orientation angle=30 deg). 143
Figure 5.13. Schematic of 8-layer laminated beam. 145
Figure 5.14. Comparison of predicted normal stresses through the thickness. 146
Figure 5.15. 3D FE model of the 8-layer laminated beam. 147
Figure 5.16. Schematic of anisotropic I section. 149
Figure 5.17. Comparison of the twist angle and twist rate. 151
Figure 5.18. Isometric view of fundamental warping modes of the anisotropic I section under unit sectional loads. 152
Figure 5.19. Planar view of fundamental warping modes of the anisotropic I section under unit sectional loads. 153
Figure 5.20. Schematic of a realistic wind turbine blade section. 154
Figure 5.21. Isometric view of fundamental warping modes of the realistic wind turbine blade section under unit sectional loads. 157